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from IPython.display import YouTubeVideo
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# WATCH THE VIDEO IN FULL-SCREEN MODE
YouTubeVideo("JXJQYpgFAyc",width=640,height=360) # Numerical integration
Question 1: Write a function that uses the rectangle rule to integrates $f(x) = \sin(x)$ from $x_{beg}= 0$ to $x_{end} = \pi$ by taking $N_{step}$ equal-sized steps $\Delta x = \frac{x_{beg} - x_{end}}{N_{step}}$. Allow $N_{step}$ and the beginning and ending of the range to be defined by user-set parameters. For values of $N_{step} = 10, 100$, and $1000$, how close are you to the true answer? (In other words, calculate the fractional error as defined below.)
Note 1: $\int_{0}^{\pi} \sin(x) dx = \left. -\cos(x) \right|_0^\pi = 2$
Note 2: The "error" is defined as $\epsilon = |\frac{I - T}{T}|$, where I is the integrated answer, T is the true (i.e., analytic) answer, and the vertical bars denote that you take the absolute value. This gives you the fractional difference between your integrated answer and the true answer.
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# Put your code here
import math
Nstep = 10
begin = 0.0
end = 3.1415926
dx = (end-begin)/Nstep
sum = 0.0
xpos = 0.0
for i in range(Nstep):
thisval = math.sin(xpos)*dx
sum += thisval
xpos += dx
error = abs(sum-2.0)/2.0
print("for dx = {0:3f} we get an answer of {1:3f} and a fractional error of {2:4e}".format(dx,sum,error))
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# WATCH THE VIDEO IN FULL-SCREEN MODE
YouTubeVideo("b0K8LiHyrBg",width=640,height=360) # Numerical differentiation
Question 2: Write a function that calculates the derivative of $f(x) = e^{-2x}$ at several points between -3.0 and 3.0, using two points that are a distance $\Delta x$ from the point, x, where we want the value of the derivative. Calculate the difference between this value and the answer to the analytic solution, $\frac{df}{dx} = -2 e^{-2x}$, for $\Delta x$ = 0.1, 0.01 and 0.001 (in other words, calculate the error as defined above).
Hint: use np.linspace() to create a range of values of x that are regularly-spaced, create functions that correspond to $f(x)$ and $\frac{df}{dx}$, and use numpy to calculate the derivatives and the error. Note that if x is a numpy array, a function f(x) that returns a value will also be a numpy array. In other words, the function:
def f(x):
return np.exp(-2.0*x)will return an array of values corresponding to the function $f(x)$ defined above if given an array of x values.
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# Put your code here
import numpy as np
%matplotlib inline
import matplotlib.pyplot as plt
def f(x):
return np.exp(-2.0*x)
def dfdx(x):
return -2.0*np.exp(-2.0*x)
x = np.linspace(-3.0,3.0, 100)
dx = 1.0e-2
deriv = (f(x+dx)-f(x-dx))/(2.0*dx)
error = np.abs((deriv-dfdx(x))/dfdx(x))
plt.plot(x,error)
print("the average fractional error is:", error.mean())
Question 3: For the two programs above, approximately how much does the error go down as you change the step size $\Delta x$ by factors of 10?
// put your answer here
The error should go down by approximately a factor of 10 as the step size goes down by a factor of 10, for both the numerical integration and differentiation.
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from IPython.display import HTML
HTML(
"""
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